An Overview of the Balanced Excited Random Walk

Abstract

The balanced excited random walk, introduced by Benjamini, Kozma and Schapira in 2011, is defined as a discrete time stochastic process in $\mathbb Z^d$, depending on two integer parameters 1 ≤ d ₁, d ₂ ≤ d, which whenever it is at a site $x\in \mathbb Z^d$ at time n, it jumps to x ± e _i with uniform probability, where e ₁, …, e _d are the canonical vectors, for 1 ≤ i ≤ d ₁, if the site x was visited for the first time at time n, while it jumps to x ± e _i with uniform probability, for 1 + d − d ₂ ≤ i ≤ d, if the site x was already visited before time n. Here we give an overview of this model when d ₁ + d ₂ = d and introduce and study the cases when d ₁ + d ₂ > d. In particular, we prove that for all the cases d ≥ 5 and most cases d = 4, the balanced excited random walk is transient.